One portion of my research over the past year that relates to this project is described in a project of Dr. Shyamal Peddada, entitled 'Statistical Methods with Applications to Toxicology and Microarray Data' (Z01-ES-101744). The majority of my research on this project, however, involved working on the development of new methods in three areas: (1) nonparametric hazard analysis with missing cause-of-death data, (2) inference about shape-constrained hazard functions, and (3) accounting for body weight in causal inference about tumor incidence. Work in the first area is based on nonparametric kernel smoothing techniques, while the other two methods were developed in a Bayesian framework and use Markov Chain Monte Carlo (MCMC) computational techniques. These three areas of research are described in more detail below.[unreadable] [unreadable] Area 1:[unreadable] [unreadable] Survival data are typically subject to censoring, such as when a study ends before all participants die. Additionally, when multiple causes of death are operating, the time to death from one cause can be censored by a death from another cause. In some situations, causes of death are missing for a subset of individuals, such as in a carcinogenicity study when the pathologist is not able to determine the role of every tumor in causing death.[unreadable] [unreadable] Survival analyses often focus on the hazard function associated with a given cause of death. We derived three nonparametric hazard estimators that are appropriate when some death times are right censored and some censoring indicators are missing. All three estimators enjoy certain large-sample properties such as uniform strong consistency and asymptotic normality. A simulation study showed that the proposed hazard estimators also performed well in small samples. In addition, we extended these methods to the regression setting, which allows us to evaluate the effects of various explanatory variables on the cause-specific hazard functions.[unreadable] [unreadable] We wrote two manuscripts to summarize the work in this area, one for the homogeneous estimation problem and another for the heterogeneous regression setting. The estimation paper was submitted to the Canadian Journal of Statistics and the regression paper was submitted to Biometrics. We are still waiting for the reviews.[unreadable] [unreadable] Area 2:[unreadable] [unreadable] When modeling time-to-event data, we often have prior information about the general shape of a hazard function. For example, we might believe that tumor incidence increases with age. Incorporation of such information, if valid, can substantially improve the efficiency of estimation procedures, and tests that capitalize on this structure should be more powerful than those that do not.[unreadable] [unreadable] Motivated by the availability of prior information in carcinogenicity studies, we developed a Bayesian solution to the general problem of making inferences about hazard rates subject to shape constraints. Restrictions can be incorporated by choosing a prior distribution that has support only among functions satisfying the constraints. We proposed a new type of prior and a computational algorithm that is efficient and yet simple to implement for censored data. Also, our model structure should facilitate generalizations to broader applications, such as hazards that are umbrella shaped or bathtub shaped.[unreadable] [unreadable] A rough manuscript has been prepared, but further computational work is necessary before the final version will be ready for peer review.[unreadable] [unreadable] Area 3:[unreadable] [unreadable] Body weight is an important confounder in the relationship between treatment and tumor incidence. Chemical exposures can cause reductions in weight, either through toxic (but non-tumorigenic) effects or by making treated (and possibly tumorigenic) food unpalatable. Incidence rates of certain tumors are associated with body weight, though the direction of the association is not always clear; lighter animals might be at lower risk of developing tumors, yet animals can lose weight after developing a tumor. Thus, a change in tumor incidence might be due to a direct tumorigenic effect of treatment, an indirect treatment-modified effect of weight, or both. Complicating the problem further is the fact that most tumors are not observable until death; and weight, tumor status, and treatment may all affect survival. Investigators are interested in assessing direct causal effects of treatment on tumor incidence through pathways not involving weight and survival.[unreadable] [unreadable] We developed a Bayesian approach that superimposes a flexible growth model on a multistate model involving a tumor state, a moribund state, and a death state. Animal-specific growth curves vary with age, tumor status, and moribund status; tumor incidence rates depend on age and time-varying body weight; and the risk of becoming moribund is a function of age, weight, and tumor status. Once moribund, we assume the risk of death is not affected by age, tumor status, or body weight history. Not only are the growth rates allowed to vary among animals differentially with age, but the slope of the growth curve can change after an animal develops a tumor or becomes moribund. Covariates can be incorporated as well. The resulting model is complex, but it should allow us to distinguish between direct treatment effects on tumor incidence and treatment-modified weight effects on tumor incidence.[unreadable] [unreadable] A manuscript has been prepared, but the associated computer program is not complete. Some fine tuning of the prior distributions and some further programming work is required before the manuscript will be ready for submission.